On the convergence rate of the Dirichlet-Neumann iteration for unsteady thermal fluid structure interaction
This work provides a theoretical convergence analysis for a common partitioned method in thermal FSI, which is useful for practitioners but is limited to simplified 1D linear cases.
The paper analyzes the convergence rate of the Dirichlet-Neumann iteration for partitioned simulation of thermal fluid-structure interaction, deriving an exact formula for the spectral radius of the iteration matrix for 1D linear heat equations. The analysis shows that convergence rate depends on aspect ratio and heat conductivity quotient, with smaller time steps improving convergence.
We consider the Dirichlet-Neumann iteration for partitioned simulation of thermal fluid-structure interaction, also called conjugate heat transfer. We analyze its convergence rate for two coupled fully discretized 1D linear heat equations with jumps in the material coefficients across these. These are discretized using implicit Euler in time, a finite element method on one domain, a finite volume method on the other one and variable aspect ratio. We provide an exact formula for the spectral radius of the iteration matrix. This shows that for large time steps, the convergence rate is the aspect ratio times the quotient of heat conductivities and that decreasing the time step will improve the convergence rate. Numerical results confirm the analysis and show that the 1D formula is a good estimator in 2D and even for nonlinear thermal FSI applications.